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Log-linear Poisson autoregression

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  • Fokianos, Konstantinos
  • Tjøstheim, Dag

Abstract

We consider a log-linear model for time series of counts. This type of model provides a framework where both negative and positive association can be taken into account. In addition time dependent covariates are accommodated in a straightforward way. We study its probabilistic properties and maximum likelihood estimation. It is shown that a perturbed version of the process is geometrically ergodic, and, under some conditions, it approaches the non-perturbed version. In addition, it is proved that the maximum likelihood estimator of the vector of unknown parameters is asymptotically normal with a covariance matrix that can be consistently estimated. The results are based on minimal assumptions and can be extended to the case of log-linear regression with continuous exogenous variables. The theory is applied to aggregated financial transaction time series. In particular, we discover positive association between the number of transactions and the volatility process of a certain stock.

Suggested Citation

  • Fokianos, Konstantinos & Tjøstheim, Dag, 2011. "Log-linear Poisson autoregression," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 563-578, March.
  • Handle: RePEc:eee:jmvana:v:102:y:2011:i:3:p:563-578
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    References listed on IDEAS

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    1. Fokianos, Konstantinos & Rahbek, Anders & Tjøstheim, Dag, 2009. "Poisson Autoregression," Journal of the American Statistical Association, American Statistical Association, pages 1430-1439.
    2. Jung, Robert C. & Kukuk, Martin & Liesenfeld, Roman, 2006. "Time series of count data: modeling, estimation and diagnostics," Computational Statistics & Data Analysis, Elsevier, pages 2350-2364.
    3. René Ferland & Alain Latour & Driss Oraichi, 2006. "Integer-Valued GARCH Process," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(6), pages 923-942, November.
    4. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, pages 307-327.
    5. Jensen, S ren Tolver & Rahbek, Anders, 2004. "Asymptotic Inference For Nonstationary Garch," Econometric Theory, Cambridge University Press, vol. 20(06), pages 1203-1226, December.
    6. Konstantinos Fokianos & Benjamin Kedem, 2004. "Partial Likelihood Inference For Time Series Following Generalized Linear Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(2), pages 173-197, March.
    7. Yunwei Cui & Robert Lund, 2009. "A new look at time series of counts," Biometrika, Biometrika Trust, pages 781-792.
    8. Richard A. Davis, 2003. "Observation-driven models for Poisson counts," Biometrika, Biometrika Trust, pages 777-790.
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    Citations

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    Cited by:

    1. Aknouche, Abdelhakim & Bendjeddou, Sara, 2016. "Negative binomial quasi-likelihood inference for general integer-valued time series models," MPRA Paper 76574, University Library of Munich, Germany, revised 03 Feb 2017.
    2. Douc, R. & Doukhan, P. & Moulines, E., 2013. "Ergodicity of observation-driven time series models and consistency of the maximum likelihood estimator," Stochastic Processes and their Applications, Elsevier, pages 2620-2647.
    3. Doukhan, Paul & Fokianos, Konstantinos & Tjøstheim, Dag, 2012. "On weak dependence conditions for Poisson autoregressions," Statistics & Probability Letters, Elsevier, pages 942-948.
    4. Moysiadis, Theodoros & Fokianos, Konstantinos, 2014. "On binary and categorical time series models with feedback," Journal of Multivariate Analysis, Elsevier, pages 209-228.
    5. Stella Kitromilidou & Konstantinos Fokianos, 2016. "Mallows’ quasi-likelihood estimation for log-linear Poisson autoregressions," Statistical Inference for Stochastic Processes, Springer, vol. 19(3), pages 337-361, October.
    6. Dag Tjøstheim, 2012. "Some recent theory for autoregressive count time series," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, pages 413-438.
    7. Konstantinos Fokianos & Dag Tjøstheim, 2012. "Nonlinear Poisson autoregression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, pages 1205-1225.
    8. Anne Leucht & Michael Neumann, 2013. "Degenerate $$U$$ - and $$V$$ -statistics under ergodicity: asymptotics, bootstrap and applications in statistics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, pages 349-386.
    9. Fokianos, Konstantinos & Moysiadis, Theodoros, 2017. "Binary time series models driven by a latent process," Econometrics and Statistics, Elsevier, vol. 2(C), pages 117-130.
    10. Aknouche, Abdelhakim & Bentarzi, Wissam & Demouche, Nacer, 2017. "On periodic ergodicity of a general periodic mixed Poisson autoregression," MPRA Paper 79650, University Library of Munich, Germany.
    11. Dag Tjøstheim, 2012. "Rejoinder on: Some recent theory for autoregressive count time series," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, pages 469-476.
    12. Fukang Zhu & Lei Shi & Shuangzhe Liu, 2015. "Influence diagnostics in log-linear integer-valued GARCH models," AStA Advances in Statistical Analysis, Springer;German Statistical Society, pages 311-335.
    13. Ali Ahmad & Christian Francq, 2016. "Poisson QMLE of Count Time Series Models," Journal of Time Series Analysis, Wiley Blackwell, pages 291-314.
    14. Doukhan, Paul & Fokianos, Konstantinos & Li, Xiaoyin, 2012. "On weak dependence conditions: The case of discrete valued processes," Statistics & Probability Letters, Elsevier, pages 1941-1948.
    15. Pedro H. C. Sant’Anna, 2017. "Testing for Uncorrelated Residuals in Dynamic Count Models With an Application to Corporate Bankruptcy," Journal of Business & Economic Statistics, Taylor & Francis Journals, pages 349-358.

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